An interpolant in predicate G\"odel logic

TL;DR

This paper investigates whether predicate G"odel logic satisfies the interpolation property, showing that a known counterexample for intuitionistic logic does admit an interpolant in G"odel logic, suggesting it may indeed have this property.

Contribution

The paper demonstrates that a counterexample for intuitionistic logic's lack of interpolation also admits an interpolant in predicate G"odel logic, providing evidence it might satisfy interpolation.

Findings

Counterexample admits an interpolant in G"odel logic

Supports the belief that G"odel logic satisfies interpolation

Uses semantic analysis to explore the property

Abstract

A logic satisfies the interpolation property provided that whenever a formula {\Delta} is a consequence of another formula {\Gamma}, then this is witnessed by a formula {\Theta} which only refers to the language common to {\Gamma} and {\Delta}. That is, the relational (and functional) symbols occurring in {\Theta} occur in both {\Gamma} and {\Delta}, {\Gamma} has {\Theta} as a consequence, and {\Theta} has {\Delta} as a consequence. Both classical and intuitionistic predicate logic have the interpolation property, but it is a long open problem which intermediate predicate logics enjoy it. In 2013 Mints, Olkhovikov, and Urquhart showed that constant domain intuitionistic logic does not have the interpolation property, while leaving open whether predicate G\"odel logic does. In this short note, we show that their counterexample for constant domain intuitionistic logic does admit an interpolant in predicate G\"odel logic. While this has no impact on settling the question for predicate G\"odel logic, it lends some credence to a common belief that it does satisfy interpolation. Also, our method is based on an analysis of the semantic tools of Olkhovikov and it is our hope that this might eventually be useful in settling this question.